The life of a recirculating pump follows a Weibull...

The life of a recirculating pump follows a Weibull distribution with parameters $\beta=2$ and $\delta=700$ hours. Determine for parts (a) and (b):
(a) Mean life of a pump
(b) Variance of the life of a pump
(c) What is the probability that a pump will last longer than its mean?

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Key Concepts

Weibull Distribution

The Weibull distribution is a continuous probability distribution widely used in reliability engineering and failure analysis to model lifetimes. It is characterized by a shape parameter (commonly denoted by beta) and a scale parameter (often denoted by delta), which together determine the behavior of the failure or survival rate.

Mean of Weibull Distribution

The mean or expected lifetime for a Weibull distribution is calculated using the formula E[T] = delta * Gamma(1 + 1/beta), where Gamma denotes the gamma function. This formulation provides the average time until failure based on the given shape and scale parameters of the distribution.

Variance of Weibull Distribution

The variance of the Weibull distribution, which measures the spread of the lifetime data, is computed using the equation Var[T] = delta^2 * (Gamma(1 + 2/beta) - (Gamma(1 + 1/beta))^2). This helps in understanding the variability around the mean lifetime.

Survival Function and Probability Beyond the Mean

The survival function in the context of the Weibull distribution gives the probability that a component lasts longer than a particular time. It is often used to determine the probability of survival beyond the mean lifetime, which is key to assessing the reliability of a component.

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